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Find lim(xto0) [x]((e^(1//x)-1)/(e^(1//x...

Find `lim_(xto0) [x]((e^(1//x)-1)/(e^(1//x)+1)),` (where `[.]` represents the greatest integer funciton).

Text Solution

Verified by Experts

`underset(xto0)lim[x]((e^(1//x)-1)/(e^(1//x)+1))=underset(hto0)lim[h]((e^(1//h)-1)/(e^(1//h)+1))`
`=underset(hto0)lim[h]((1-e^(-1//h))/(1+e^(-1//h)))`
=0xx(1-0)/(1+0)=0
`underset(xto0)lim[x]((e^(1//x)-1)/(e^(1//x)+1))=underset(hto0)lim[-h]((e^(-1//h)-1)/(e^(-1//h)+1))`
`=(-1)xx(0-1)/(0+1)=1`
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